Deep neural networks architectures from the perspective of manifold learning

Despite significant advances in the field of deep learning in applications to various areas, an explanation of the learning process of neural network models remains an important open question. The purpose of this paper is a comprehensive comparison and description of neural network architectures in terms of geometry and topology. We focus on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of a data manifold on different layers. In this paper, we use the concepts of topological data analysis (TDA) and persistent homological fractal dimension. We present a wide range of experiments with various datasets and configurations of convolutional neural network (CNNs) architectures and transformers in CV and NLP tasks. Our work is a contribution to the development of the important field of explainable and interpretable AI within the framework of geometrical deep learning.